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Derivative Error and Lagrange

  1. Feb 21, 2007 #1
    Consider the function f(t) = ln (1 +2x)

    Give a formula for f^(n) (x) [**the nth derivative] valid for all n >= 1 and find an upper bound for | f^(n) (x) | on the interval -0.25 <= x <= 0.25.
    [ the error ].

    I found the nth derivative to be

    f^(n) (x) = (-1)^n+1 * 2^n /n * n!
    --------------------------------
    (1 + 2x)^n

    so for
    first derivative = 2 / 1+2x
    second " " = -4 / (1+2x)^2
    third " " = 16 / (1+2x)^3
    etc.

    now for the error i kno there is a lagrange error bound equation for taylor polynomials, but the question isnt for the taylor polynomial, only the "derivative generator"

    i kno the max |f^(n+1)| <= M on an interval

    so i just need help dealing with only the derivative error and i also want to kno how to find the M value in general with Taylor polynomials (not part of the above question)

    where |f(x) - P_n(x)| <= M / (n+1)! * |x-a|^(n+1)
    for interval between a and x
     
  2. jcsd
  3. Feb 21, 2007 #2
    You are essentially done when you minimize the denominator in absolute value. Note, only "an upper bound" is asked for: You have room to over estimate.
     
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